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Li & Hitt/Price Effects in Online Product Reviews
RESEARCH ARTICLE
PRICE EFFECTS IN ONLINE PRODUCT REVIEWS: ANANALYTICAL MODEL AND EMPIRICAL ANALYSIS1
By: Xinxin Li
School of Business
University of Connecticut
2100 Hillside Road U1041
Storrs, CT 06279
U.S.A.
Lorin M. Hitt
The Wharton School
University of Pennsylvania
3730 Walnut Street, 500 JMHH
Philadelphia, PA 19104-6381
U.S.A.
Abstract
Consumer reviews may reflect not only perceived quality but
also the difference between quality and price (perceived
value). In markets where product prices change frequently,
these price-influenced reviews may be biased as a signal of
product quality when used by consumers possessing no
knowledge of historical prices. In this paper, we develop an
analytical model that examines the impact of price-influenced
reviews on firm optimal pricing and consumer welfare. We
quantify the price effects in consumer reviews for different
formats of review systems using actual market prices and on-
1Chris Kemerer was the accepting senior editor for this paper. Mike Smith
served as the associate editor.
The appendices for this paper are located in the Online Supplements
section of theMIS Quarterlys website (http://www.misq.org).
line consumer ratings data collected for the digital camera
market. Our empirical results suggest that unidimensional
ratings, commonly used in most review systems, can besubstantially biased by price effects. In fact, unidimensional
ratings are more closely correlated with ratings of product
value than ratings of product quality. Our findings suggest
the importance for firms to account for these price effects in
their overall marketing strategy and suggest that review
systems could better serve consumers by explicitly expanding
review dimensions to separate perceived value and perceived
quality.
Keywords: Online product reviews, review bias, price
effects, empirical analysis, optimal pricing
Introduction
In recent years, there has been growing research interest in
examining dissemination of product information through
online word-of-mouth. Consumers share product evaluations
of a wide assortment of products through product review web-
sites, discussion forums, blogs, and virtual communities.
These networks serve many of the same functions as tradi-
tional word-of-mouth communications (Godes et al. 2005)
that previously occurred only among family or friends. The
large-scale experience-sharing among consumers in these
networks potentially reduces uncertainty about the quality of
products or services that cannot be inspected before purchase
and therefore can play a substantial role in consumers pur-
chase decision processes. According to a survey by Deloittes
Consumer Products group (Deloitte 2007), almost two-thirds
of consumers read consumer-written product reviews on the
Internet. Among those consumers who read reviews, 82 per-
cent say their purchase decisions have been directly influ-
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Li & Hitt/Price Effects in Online Product Reviews
enced by the reviews and 69 percent share the reviews with
friends, family, or colleagues, thus amplifying their impact.
Other consulting reports and surveys have also shown that for
some consumers and products, consumer-generated reviews
are more valuable than expert reviews (ComScore 2007; Piller
1999), have a greater influence on purchasing decisions than
traditional media (DoubleClick 2004), and have a significantimpact on offline purchase behavior (ComScore 2007).
Information is being exchanged online in unprecedented scale
and detail. This increased transparency makes it possible for
researchers to monitor word-of-mouth over time and, there-
fore, to obtain a deeper understanding of consumer pre-
ferences and decision processes. The predominant research
focus has been on the correlation between consumer reviews
and sales (Chen et al. 2007; Chevalier and Mayzlin 2006;
Clemons et al. 2006; Dellarocas et al. 2004; Duan et al. 2008;
Forman et al. 2008; Godes and Mayzlin 2004). How well
these review forums communicate product information,
however, has been less studied. Li and Hitt (2008) found thatbecause early adopters of products have different preferences
for the underlying products, the reviews provided by these
early adopters are not necessarily representative of the market
as a whole. In addition, consumers do not appear to account
for this review bias when they utilize these reviews to assist
their purchase decisions. These findings suggest the impor-
tance of understanding the process by which consumers
utilize review information, the process by which consumers
decide to produce reviews, and the information that con-
sumers convey through reviews. Although the former two
issues are probably more amenable to laboratory study, the
information content of reviews can be examined through a
natural experiment present on the Internet. In particular, weexamine in this study whether product price influences
consumer reviews, and how firms can optimize their pricing
strategy to take account of price effects in consumer reviews.
There are two reasons why price may play a role in consumer
reviews. First, consumer reviews may reflect not only the
perceived quality of a product or service but also the
perceived valuethe difference between the utility derived
from product quality and pricefrom the purchase. Price has
been shown to be a major influence on customer satisfaction
in the manufacturing industry as a whole (Tsai 2007), as well
as in service industries such as rental cars (McGregor et al.
2007). There is also anecdotal evidence that consumers takeprice into account when they write reviews. For example, in
a review written on December 28, 2005, for the SONY Cyber
Shot DSC-S40 digital camera posted on CNet.com, a
consumer writes some problems but at this price cant com-
plain .But for a 4 Mp camera at this price it is fantastic!
and gives a rating of 8 (out of 10) while for the same camera
the CNet editor gives a rating of 6.6 (out of 10). Similar
observations can be found in reviews for other products.
Second, previous research suggests that, when faced with
quality uncertainty, consumers are likely to use price as a
signal of quality (Dodds et al. 1991; Grewal 1995; Kirmani
and Rao 2000; Mitra 1995; Olson 1997; Rao and Monroe
1988, 1989). Some of this quality expectation may be dis-
confirmed by actual experience. Because disconfirmation of
prior expectations is known to influence satisfaction (Cadotteet al. 1987; Churchill and Surprenant 1982; Spreng et al.
1996; Rust et al. 1999), price may have an indirect effect on
perceived quality, which is ultimately reflected in consumer
reviews.
In this study, we first develop an analytical model of optimal
firm pricing that accounts for the possibility that prices have
an indirect effect on demand by altering consumer reviews.
Next, to validate the assumptions of our model, we use actual
market prices data and online consumer ratings data to
empirically examine whether and to what extent price affects
consumer ratings. Finally, we further validate our theoretical
model predictions by examining whether the pricequalityrelationship anticipated in our theoretical model appears in the
data.
Our empirical analysis on how prices affect reviews is done
by comparing different review systems that cover the same
product. Different review systems have different methods for
collecting and displaying consumer review information. Most
of these systems are one-dimensional, having only a single
overall rating (e.g., CNet.com or Amazon.com for most pro-
ducts). Other systems divide ratings into multiple categories.
For example, Dpreview.com, an online consumer review
service focusing on digital cameras, divides consumer reviews
into five categories: construction, features, image quality,ease of use, and value for money. Among them, the value for
money dimension directly captures the role of price in ratings.
For review systems with one-dimensional ratings, there is no
explicit value for money component, but the influence of
price may be directly incorporated into the single rating. By
examining the relationship between reviews and prices over
time and across different review systems, we can investigate
the role of price in consumer valuation and product ratings.
In particular, in this paper, we empirically examine the
following assumptions and predictions of our analytical
model:
(1) Does price affect consumer reviews?(2) How is the role of price in consumer reviews different
across different review systems?
(3) Are observed prices consistent with our model of how
firms should respond to price effects in reviews?
Our empirical and analytical findings apply especially to mar-
kets where the product price changes frequently and at least
some consumers are unable or unwilling (perhaps due to high
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Li & Hitt/Price Effects in Online Product Reviews
search costs) to seek historical prices to make proper adjust-
ments. Understanding the role of price in affecting consumer
reviews is potentially useful for websites designing review
services to improve the efficiency of reviews in signaling
product quality, and it is useful for firms attempting to
understand the feedback provided by the market from product
review sites to develop optimal pricing. For instance, ourresults suggest that firms can boost ratings of their products
at release via low introductory pricing. Compared to other
strategies for managing product reviews, such as hiring paid
reviewers to create artificially high ratings (Dellarocas 2006;
Mayzlin 2006), pricing strategically to influence consumer
satisfaction at early stages of a product life cycle may be more
cost-effective and less subject to ethical concerns.
Literature Review
Since Rogers (1962), word-of-mouth has been perceived as animportant driver of sales in the product diffusion literature.
Those models normally assume that consumers experience
with a product is communicated positively through word-of-
mouth (Mahajan and Muller 1979) and therefore facilitates
product diffusion. Prior empirical work on the relationship
between word-of-mouth and product adoption usually
measures the presence of word-of-mouth by inferring its
existence and impact based on the opportunity for social
contagion instead of observing it directly (Godes et al. 2005).
For example, Reingen et al. (1984) infer word-of-mouth
interaction based on whether individuals live together, and
Foster and Rosenzweig (1995) infer knowledge spillover
through word-of-mouth based on whether farmers live in thesame village.
The emergence of large-scale online communication networks
provides a channel for researchers to directly observe word-
of-mouth over time and therefore to obtain a deeper under-
standing of consumer preferences and decision processes.
Based on product reviews or conversation data collected from
consumer networks, researchers are able to directly test the
relationship between word-of-mouth and product sales in
different industries. In the book industry, Chevalier and
Mayzlin (2006) demonstrated that the differences between
consumer reviews posted on the Barnes & Noble site and
those posted on Amazon.com were positively related to thedifferences in book sales on the two websites. Two recent
studies further found that reviews written by consumers from
the same geographic location (Forman et al. 2008) or with a
higher helpfulness vote (Chen et al. 2007) have a higher
impact on sales. In the motion picture and television indus-
tries, Godes and Mayzlin (2004) showed a strong relationship
between the popularity of a television show and the
dispersion of conversations about TV shows across online
consumer communities. Dellarocas et al. (2004) incorporated
the sentiment of word-of-mouth into a product diffusion
model and found that the valence (average numerical rating)
of online consumer reviews is a better predictor of future
movie revenues than other measures they considered. In con-
trast, Duan et al. (2008) suggested that the alternative causal
relationship is also true: that the number of online reviewsinfluences box office sales. In the beer industry, Clemons et
al. (2006) found that the variance of ratings and the strength
of the most positive quartile of reviews have a significant
impact on the growth of craft beers.
Although the link between word-of-mouth and product sales
was generally supported in the aforementioned studies, other
researchers began to further examine this relationship by
examining whether online reviews are effective in communi-
cating actual product quality. Anderson (1998) proposed that
consumers are more likely to engage in word-of-mouth when
they have extreme opinions. Bowman and Narayandas (2001)
further suggested that word-of-mouth behavior is also drivenby customer loyalty to the brand. Li and Hitt (2008) found
that consumers self-selection behavior in purchasing may
introduce bias into consumer reviews, which further affects
sales and consumer welfare.
Built on the relationship between word-of-mouth and product
sales, analytical studies of online reviews further examined
how firms profitability and marketing strategies can be
affected. Dellarocas (2006) and Mayzlin (2006) examined the
incentive of firms to manipulate reviews and the implications
of this to consumer welfare. Chen and Xie (2008) showed
that firms only have incentive to help disseminate consumer
reviews when the firms target market is sufficiently large,and that reviews affect the optimal product assortment and
information provision policies of firms. Li et al. (2010) found
that an S-shaped relationship exists between the quality of
reviews and firm profits.
In all these existing studies, it is implicitly assumed that
consumer reviews reflect consumers perceptions of product
quality and hence consumers are not affected by price.
Whether this assumption is consistent with consumer behavior
is left untested.
Zeithaml (1988, p. 14) defines value as the consumers over-
all assessment of the utility of a product based on perceptionsof what is received and what is given. Value implicitly refers
to a buyers trade-off between benefit (quality) and cost
(price) of the purchase (Bolton and Drew 1991). Although it
is generally agreed that a consumers purchase decision is
determined by expected utility before purchase, perceived
value from a purchase can also affect the consumers ex post
satisfaction with the purchase (Spreng et al. 1996). Corre-
spondingly, price, in addition to quality, should also be an
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Li & Hitt/Price Effects in Online Product Reviews
important factor influencing consumers post purchase
satisfaction (Varki and Colgate 2001; Voss et al. 1998).
Richins (1983) suggested that word-of-mouth may be driven
by consumer satisfaction with a purchase. Therefore price, in
turn, should influence consumer reviews.
A distinct line of argument suggests that consumers may useprice as a signal of quality before they make purchase
decisions when facing quality uncertainty ( Dodds et al. 1991;
Grewal 1995; Kirmani and Rao 2000; Mitra 1995; Olson
1997; Rao and Monroe 1988, 1989). Many studies have
shown that consumers post purchase satisfaction can be
affected by the confirmation or disconfirmation of received
quality after consuming the product versus their expectation
before purchase (Cadotte et al. 1987; Churchill and Surpre-
nant 1982; Rust et al. 1999; Spreng et al. 1996). Accordingly,
by influencing consumers expectations over quality before
purchase, price may indirectly affect post-purchase consumer
satisfaction and further affect consumer ratings.
In the next section, we first construct an analytical model that
examines how firms can optimally price a product when price
can have an effect on consumer reviews. In a later section,
we empirically test our model assumptions and predictions
using actual market prices data and online consumer ratings
data collected for the digital camera market.
Analytical Model
In this section, we first examine firms optimal pricing stra-
tegy in a monopoly setting and discuss the implications of theprice effects in consumer reviews for consumer surplus. We
then further verify the generalizability of our model predic-
tions to duopoly settings.
Model Setup
We consider a two-period market for an experience good in
which, in each period, a group of consumers comes into the
market and makes a decision about whether to purchase up to
one unit of the durable good with no repeat purchase oppor-
tunity. As suggested by standard product diffusion modeling
approaches (e.g., Bass 1969), early adopters rely primarily ontheir own expectations, whereas later adopters can also be
influenced by peer opinions such as consumer reviews.
The net utility of consuming the product for consumer i is
defined as U (xi, q,p) = qp txi, wherepis the price of the
product, which may vary over time. The value of element q
measures the objective quality of the product and is the
same for all consumers. To capture uncertainty in the quality
of a product prior to purchase, qis a random draw from the
interval [0, 1]. Consumers learn the actual value of qonly
after buying the product. To allow for horizontal differentia-
tion in preferences for observable product attributes (e.g.,
color), we introduce a taste parameter (xi), which represents
the position of a consumers ideal product in a product space
(a bounded interval [0, 1]). The actual product occupies posi-
tion 0 and this position is assumed exogenous (results aresimilar if we assume the actual product occupies position
instead). Consumers know the value of xibefore purchase
and reduce their utility by a factor tper unit distance from the
product, analogous to the travel distance in the Hotelling
model (Hotelling 1929). We assume t> 1 so that, when com-
bined with the fact that q #1, a monopoly cannot cover the
whole market.
In the first period, early adopters arrive and make their pur-
chase decisions based on their expectation over quality (qe)
and first-period price (p1). qeis exogenously given and com-
mon across all consumers.2 Without loss of generality, we
normalize the value of the best alternative to this product to
be zero. Thus, only consumers with expected utility U (xi, qe,
p1) larger than zero will buy the product. The first period
demand equals (qe p1)/t. After consumers experience the
product, they post their evaluations (R) online; the evaluations
can be accessed by consumers who arrive in the second
period. The second-period consumers will always wait for
reviews to be posted before making purchase decisions. They
update their quality perceptions based on these reviews,
which affect their expected utility and therefore affect the
second-period demand.
The distinctive feature of our model is that we allow reviews(i.e., reviewers post purchase evaluations of the product) to
be affected by both product quality as well as by the prices
that reviewers paid. This assumption will be empirically
tested using actual prices and ratings data (see the Empirical
Analysis section). We normalize our rating measure such
that it is numerically comparable to quality (bounded in the
range [0, 1]), withRbeing equal to Max{0, Min{1, q b(p1 r(q))}}. The parameter bcaptures the strength of the priceeffects (0 < b< 1) and r(q) captures the market price for a
product of quality qthat is perceived as reasonable by con-
sumers in the sense that deviations from this price have addi-
2Exogenous prior expectation assumption has appeared in prior studies on
pricing of experience goods (e.g., Schmalensee 1982; Shapiro 1983a, 1983b;
Villas-Boas 2004). Shapiro (1983b) points out that consumers expectations
about a new products quality are generally not fully rational. Many factors
can affect a consumers expectation over a products quality before purchase,
such as advertising, prior experience with the brand, and average quality level
of similar products in the market. If qediffers across consumers, then we can
include q ieinxi, and the subsequent analysis still applies.
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Li & Hitt/Price Effects in Online Product Reviews
Figure 1. Sample Review Pages from Dpreview.com
tional effects on value perception. Thus, products with exces-
sively high prices have their ratings (R) reduced by bper unit
of price above the nominal level r(q), while products that are
less expensive receive a boost in their ratings of b (r(q) p1).
To provide some structure to r(q), we assume r(q) equals the
standard monopoly price for products with quality q: r(q) =
q/2. Our results do not appear to be sensitive to this assump-
tion as long as r(q) is an increasing function of q.3
In the second period, the followers arrive in the market and
use reviews posted on different websites to form expectations
of product quality before purchase. Depending on the type of
websites they visit and whether they are willing to spend
effort to seek historical price information, their expectations
of quality may be influenced by reviews differently.
For our purposes, it is useful to divide consumer review
systems into two types: those that provide only a single rating,
and those that provide multiple rating dimensions. We are
interested in comparing single rating systems to multidimen-
sional rating systems that include a direct measure of consu-
mers value perceptions (the total utility of quality less price).
Figure 1 shows a rating page from Dpreview.com, which has
multidimensional ratings; Figure 2 shows a rating page from
CNet.com, which utilizes a single review dimension.
3We also tried the assumption of r(q) being a nonlinear function of q, which
also turnsRinto a nonlinear function of q, and obtained very similar results.
Derivation is available upon request from the authors.
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Li & Hitt/Price Effects in Online Product Reviews
Figure 2. Sample User Opinions Page from CNet.com
Consumers who read reviews from websites that separate
quality ratings from value ratings can derive qdirectly from
average ratingReven without knowledge of historical price.
However, consumers who read reviews from websites that
integrate product quality and influence of price into unidimen-
sional ratings may not be aware of or are not able to derive q
from R unless they are willing to incur a cost to retrieve
historical price information.4 The net effect of these behav-
iors, similar in spirit to search cost models, is that some frac-
tion of consumers a (0 < a< 1) will be partially uninformed
in the sense that they set their quality estimate to a value
implied by the reviews they observe (R), while the remainder
of consumers are fully informed and know q.5
4According to web browsing behavior tracked by ComScore in January 2006,
more than 10 times as many camera buyers visited review websites with
unidimensional ratings as those who visited review sites with multidimen-
sional ratings (of the rating sites we consider), and 2.5 times as many camera
buyers purchased from retailers who provide unidimensional ratings.
Accordingly, if unidimensional ratings are indeed affected by price and are
a biased signal of quality as demonstrated in our empirical analysis, this bias
can have a significant impact on consumer purchasing behavior and thus on
consumer welfare.
5For simplicity, we assume consumers updated expectations are R (or q)
without modeling the detailed belief updating process. If, alternatively, we
assume that a consumers prior expectation has a mean qe
and a variance e2
,each review has a meanR (or q) and a variance r
2 and the number of reviews
is m, then following Bayes Rule, the updated expectation will be
(or )Rm q
m
e
e
e
2 2 2
2 2 2
+
+
qm q
m
e
e
e
2 2 2
2 2 2
+
+
(DeGroot 1970, p. 168), which is a linear function ofR(or q) and converges
to R (or q) as mincreases. Therefore, this simplified assumption will not
affect our results qualitatively.
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Li & Hitt/Price Effects in Online Product Reviews
Solid line: prices influence reviews a= 0.8, b= 0.8, qe= 0.5, and n= 3
Dotted line: no price effects
Figure 3. Comparison of Optimal Monopoly Prices, Rating, and Profit between Our Model (Prices
Influence Reviews) and the Benchmark (No Price Effects)
(4) Only firms selling high-quality products produce higher
profit (q >
_
Q3) when reviews reflect both price and
quality than when reviews are pure quality measures.6
The intuition behind this set of pricing results can be ex-
plained as follows. If product quality is very high ( ),q bq
b
e
>+
+
2
2
the firm can achieve maximal ratings without having to
discount the price at all. This is because quality uncertainty
in the first period encourages the firm to price lower than
would be optimal under full information, and this reduced
price gives consumers the perception that the product was not
only of high quality but also of high value given the price.
This boost from the price component raises reviews over what
they would be if only quality mattered, allowing the firm to
increase price above the second period benchmark level (of
no price influence in reviews).
Firms selling low-quality products face a strategic decision:
whether to exploit quality uncertainty in the first period by
charging high prices and then lose sales in the second period
from the additional negative effects of high prices on reviews,
or to price low to try to create high product reviews. When q
is very low (q
+
1
2
2,
low ( ). The dividing thresholdq Max Qq
abn
e
9, later adopters make up
most of the sample).
Although our results suggest that the price effects are
substantial at least for the product category we consider,whether similar effects can be observed for other categories
of products needs to be tested in future research. There are
some other limitations of this work. First, because we can
only observe the market-level average price data instead of
the individual-level price paid by each reviewer, we are not
able to associate each rating with the actual transaction price.
As a result, we are forced to use aggregate measures for both
ratings and prices (per-month cumulative average rating and
price) instead of each single review. This limitation may,
however, make our estimated coefficient of price more con-
17Estimates of model (3) in column I of Table 4 suggest
dCNetConsumerRatingit/ dAvgPrice
it-1= -0.71 / AvgPrice
it-1. According to
Table 1, CNet consumer ratings range from 5 to 10 and average market prices
range from $70 to $1,696. If we scale both ratings and prices to [0, 1] as
assumed in our analytical model and utilize the mean of average market
prices $435 (Table 1) as the base number on the right hand side of the
equation, we can derive that the price effects parameter (b) is around 0.53.
A sample of Comscore web visiting data (footnote 4) suggests that the
portion of consumers who visit single-dimension review websites (value of
parameter a) is above 90%.
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Li & Hitt/Price Effects in Online Product Reviews
servative, and indeed strengthen the argument that single-
value ratings may largely reflect the consumers perceived
value from purchase instead of pure quality.
Second, we do not have sales data associated with reviews
posted on Dpreview.com versus those posted on CNet.com,
and hence we are not able to empirically compare the impact
of reviews on sales across two different review formats
(single-value and multiple-value ratings). Future study with
finer-grained data may examine the indirect impact of price
on sales through influencing consumer ratings and the extent
to which the consumers purchase behavior is affected by
availability of separate quality ratings. In particular, it would
be useful to know to what extent consumers are able to
correct the bias in ratings caused by price.
Finally, in this paper, we use arrival time to separate early
adopters and later followers and assume the total market size
to be exogenous. Similar assumptions have also been used in
previous studies of consumer reviews (e.g., Chen and Xie
2008). Whether reviews can affect consumers redistribution
across periods or affect the market size can be a fruitful
direction for future study.
Acknowledgments
The authors would like to thank the Mack Center for Technology
Innovation at Wharton for funding this research and thank the NPD
Group and CIDRIS at University of Connecticut for data support.
The authors would also like to thank the senior editor, Chris F.
Kemerer, the associate editor, and the two anonymous reviewers forvaluable comments and suggestions.
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About the Authors
Xinxin Liis an assistant professor of Operations and Information
Management at the School of Business, University of Connecticut.
She received her Ph.D. from The Wharton School, University of
Pennsylvania. Her research interests lie at the intersection of infor-
mation systems and marketing. Her current research examines the
economics of online word of mouth and competition in business-to-
business and business-to-consumer markets. Her work has appeared
inInformation Systems Research.
Lorin M. Hittis the Class of 1942 Professor at the Wharton School
of the University of Pennsylvania. His work focuses on the produc-
tivity of information technology investments and the economics of
electronic business. He received his Ph.D. in Management from
MIT and his Sc.B. and Sc.M. degrees in Electrical Engineering from
Brown University. He is currently serving as the co-Departmental
Editor for Information Systems atManagement Science, and is on
the editorial board of theJournal of MIS.
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RESEARCH ARTICLE
PRICE EFFECTS IN ONLINE PRODUCT REVIEWS: ANANALYTICAL MODEL AND EMPIRICAL ANALYSIS
By: Xinxin Li
School of Business
University of Connecticut
2100 Hillside Road U1041
Storrs, CT 06279
U.S.A.
Lorin M. Hitt
The Wharton School
University of Pennsylvania
3730 Walnut Street, 500 JMHH
Philadelphia, PA 19104-6381
U.S.A.
Appendix A
Derivation of Optimal Price Functions for the Monopoly Setting
We apply backward induction to derive optimal price functions. In the second period, given first period pricep1, the firm selects second period
pricep2(p2
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The corresponding second period profit as a function ofp1is
Back in the first period, given *2(p1), the firm selects a first period price p1 (p1 < qe) to maximize its total profit in both periods:
. By comparing optimal profit in different ranges ofp1, we can derive the optimal first period price for different valuesp q p
tp
e
1 1
2 1
( )( )*
+
of q:
Combiningp*1andp*2(p1), we can derive that .
Appendix B
Derivation of Optimal Price Functions for the Duopoly Setting
We first utilize the case of q1=12 to explain in detail how to derive equilibrium prices and then follow the same procedure to solve equilibria
for the other two cases (q1= 1 and q1= 0).
We apply backward induction to derive optimal price functions. In the second period, given the first period prices, p11and p21(pj1< qej=
12),
the ratings of the two products are and . Given a= 1, b= 1, n= 3, t= 16,{ }{ }R Min Max p
1
3 4
41 0 11
=
, , { }{ }R Min Max q p
2
3 2
21 0 2 21
=
, ,and qe1 = q
e2 =
12, if all second-period consumers purchase from one of the two firms, the second period profits are
and . If some( ){ }{ }( )12 12 1 2 12 22 163 1 0 3= + +p Min Max R R p p, , ( ){ }{ }( )22 22 2 1 22 12 163 1 0 3= + +p Min Max R R p p, ,second-period consumers expect negative utility from both firms and do not buy from either firm, the profit functions are
and . Then back in the first period, firms select( ){ }{ }( )12 12 1 123 1 0 6= p Min Max R p, , ( ){ }{ }( )22 22 2 223 1 0 6= p Min Max R p, ,p11 and p21 to maximize their total profits in both periods: and{ }{ } 1 11 11 21 16 123 1 0= + + +p Min Max p p, , *
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. It can be proved that in this scenario all of the second-period consumers will purchase{ }{ } 2 21 21 11 16 223 1 0= + + +p Min Max p p, , *one of the two products in equilibrium. Thus, firms second period profit functions are:
,{ }{ }( ){ }12 12 3 44 3 22 12 22 163 1 3 1 011 2 21= + +
p Min Min Max p pp q p
, , ,
.{ }{ }( ){ }22 22
3 2
2
3 4
2 22 12
163 0 3 1 0
2 21 11
= + +
p Max Min Max p pq p p
, , ,
We can then derive the optimal second period prices as functions of the first period prices:
,
The corresponding second period profits as functions of the first period prices thus are:
,
.
Then back in the first period, firms select the first period prices to maximize their total profits in both periods:
,{ }{ } ( ) 1 11 11 21 16 12 1 1 213 1 0= + + +p Min Max p p p p, , ,*
.{ }{ } ( ) 2 21 21 11 16 22 1 1 2 13 1 0= + + +p Min Max p p p p, , ,*
By comparing profits in different ranges of p11and p21, we can derive the optimal first period prices for different values of q2:
, .
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Combiningp*11,p*21,p
*12(p11,p21), andp
*22(p11,p21), we can derive that:
, .
Following similar procedure, we can derive the optimal price functions for q1= 1:
, .
, .
Similarly, the optimal price functions for q1= 0 are
, .
, .
In the benchmark scenario, the firms selectp11,p21,p12, andp22 to maximize their total profits:
,{ }{ } ( ){ }{ }( )1 11 11 21 16 12 1 2 12 22 163 1 0 3 1 0 3= + + + + +p Min Max p p p Min Max q q p p, , , ,
.{ }{ } ( ){ }{ }( )2 21 21 11 16 22 2 1 22 12 163 1 0 3 1 0 3= + + + = + +p Min Max p p p Min Max q q p p, , , ,
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It can be shown that the optimal first period prices are both 16, the second period prices are:
(1) If .q pif q
q if qp
if q
if q
q q
1 21
16
1
312 2
5
6 2 2
1
2
22
16
1
312 2
2
1
2
11
0
1
0 0
2 2
= =
+